Pólya’s fundamental enumeration theorem and some results
from Williamson’s generalized setup of it are proved in terms of Schur-
Macdonald’s theory (S-MT) of “invariant matrices”. Given a permutation
group W ≤ Sd and a one-dimensional character χ of W , the polynomial
functor Fχ corresponding via S-MT to the induced monomial representation
Uχ = ind|Sdv/W (χ) of Sd , is studied. It turns out that the characteristic ch(Fχ )
is the weighted inventory of some set J(χ) of W -orbits in the integer-valued
hypercube [0, ∞)d . The elements of J(χ) can be distinguished among all
W -orbits by a maximum property. The identity ch(Fχ ) = ch(Uχ ) of both
characteristics is a consequence of S-MT, and is equivalent to a result of
Williamson. Pólya’s theorem can be obtained from the above identity by
the specialization χ = 1W , where 1W is the unit character of W.